May depend on the EGPWS equipment manufacturer (Airbus offers a choice); also if the EGPWS has a GPS receiver embedded in the box vice an external input (Honeywell option).
Failure of the embedded GPS would presumably give an EGPWS fault.
Dispatch without an external GPS might depend on the accuracy of the remaining navigation source to be used for the dispatch and how EGPWS uses this. Furthermore if that nav source degrades – e.g. no DME cover, then system status/alerts might be given which could affect flight operations.
Read the MEL and dispatch deviation guide (DDG) carefully and ensure that you understand any implications or restrictions.